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Theorem dia1eldmN 31853
Description: The fiducial hyperplane (the largest allowed lattice element) belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
dia1eldm.h  |-  H  =  ( LHyp `  K
)
dia1eldm.i  |-  I  =  ( ( DIsoA `  K
) `  W )
Assertion
Ref Expression
dia1eldmN  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W  e.  dom  I
)

Proof of Theorem dia1eldmN
StepHypRef Expression
1 eqid 2296 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 dia1eldm.h . . . 4  |-  H  =  ( LHyp `  K
)
31, 2lhpbase 30809 . . 3  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
43adantl 452 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W  e.  ( Base `  K ) )
5 hllat 30175 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
6 eqid 2296 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
71, 6latref 14175 . . 3  |-  ( ( K  e.  Lat  /\  W  e.  ( Base `  K ) )  ->  W ( le `  K ) W )
85, 3, 7syl2an 463 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W ( le `  K ) W )
9 dia1eldm.i . . 3  |-  I  =  ( ( DIsoA `  K
) `  W )
101, 6, 2, 9diaeldm 31848 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( W  e.  dom  I 
<->  ( W  e.  (
Base `  K )  /\  W ( le `  K ) W ) ) )
114, 8, 10mpbir2and 888 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W  e.  dom  I
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039   dom cdm 4705   ` cfv 5271   Basecbs 13164   lecple 13231   Latclat 14167   HLchlt 30162   LHypclh 30795   DIsoAcdia 31840
This theorem is referenced by:  dia1elN  31866
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-poset 14096  df-lat 14168  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-lhyp 30799  df-disoa 31841
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